Integration Study Guide

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Notation and Geometric Ideas

Identify the type of integral described by the notation.

An integral of the form $\displaystyle \int _C \textbf {F} \cdot d\textbf {r}$ is a:

Double Integral Scalar Line Integral Vector Line Integral

What does this integral represent geometrically?

Signed area Sum of components of $\textbf {F}$ along $C$ Signed volume

Does the orientation of $C$ matter for the integral?

Yes No

An integral of the form $\displaystyle \iint _D f(x,y)dA$ is a:

Double Integral Scalar Line Integral Vector Line Integral Scalar Surface Integral Vector Surface Integral

What does this integral represent geometrically?

Signed area Sum of components of $\textbf {F}$ along $C$ Signed volume

If $f(x,y) = k$ for some constant $k$ , then $\displaystyle \iint _D f(x,y)dA$ is:

$k$ times the area of $D$ $k$ times the volume of the region under $z=k$ and over $D$

The integral $\displaystyle \iint _D f(x,y)dA$ will be negative in which of the following instances (select all):

The function $f(x,y) \leq 0$ on all of $D$ $f(x,y) \geq 0$ and the region $D$ in the $xy$ -plane is in the region $y \leq 0$ The represented solid has more volume under the $xy$ -plane than above it

An integral of the form $\displaystyle \int _C f(x,y)\,ds$ is a:

Double Integral Scalar Line Integral Vector Line Integral

What does this integral represent geometrically?

Signed area between $f(x,y)$ and the curve $C$ in the $xy$ -plane Sum of components of $\textbf {F}$ along $C$ Signed volume

If $f(x,y) = 1$ , then $\displaystyle \int _C 1\,ds$ is:

The area enclosed by $C$ when $C$ is a closed curve The arc length of $C$ $1$

Does orientation of $C$ matter for this type of integral?

Yes No

An integral of the form $\displaystyle \int _C Pdx+Qdy$ is a:

Double Integral Scalar Line Integral Vector Line Integral

Computations

These questions are more open ended but are meant to guide your studying a bit.

What is your thought process when asked to evaluate $\displaystyle \int _C f(x,y)\,ds$ for some $f(x,y)$ and some $C$ ?

What is your thought process when asked to evaluate $\displaystyle \iint _D f(x,y)\,dA$ for some $f(x,y)$ and some $D$ ?

What is your thought process when asked to evaluate $\displaystyle \int _C \textbf {F} \cdot d\textbf {r}$ for some vector field $\textbf {F}$ and some $C$ ?

How do you check whether a vector field $\textbf {F}$ is conservative? How does this relate to the curl?

What does the Fundamental Theorem of Line Integrals say, and what are its main consequences for the purposes of this class?

How can the Fundamental Theorem of Line Integrals and its consequences help you compute vector line integrals?

When can you use the Fundamental Theorem of Line Integrals?

What does Green’s Theorem say? What are its conditions, and in what instances can it be used?

Can we use Green’s Theorem if $\textbf {F}$ is conservative? What does it tell us in this case?

Suppose we have a line integral $\displaystyle \int _C \textbf {F} \cdot d\textbf {r}$ . If we need to close the curve $C$ to use Green’s Theorem, what are the steps we must carry out in order to solve the original integral?

When given a vector line integral $\displaystyle \int _C \textbf {F} \cdot d\textbf {r}$ , what is your thought process in determining what method to use?